How we cite our quotes: (Book.Chapter.Paragraph)
Quote #4
In modern times, the diligence of publick teachers is more or less corrupted by circumstances, which render them more or less independent of their success and reputation in their particular professions. (5.1.77)
Adam Smith thinks that schoolteachers would perform much better if their salaries were directly connected to the demand for their services. But public teachers are paid a salary and students are forced to be in their class, so there's no incentive for the teacher to improve the way she or he does things.
Quote #5
Were there no publick institutions for education, no system, no science would be taught for which there was not some demand. (5.1.78)
Adam Smith isn't sure what to think of public education. On the one hand, he's a fan of publicly accessible education. But on the other hand, he believes that education should be governed by supply and demand and that teachers should only teach what people want to learn about.
Quote #6
A private teacher could never find his account in teaching, either an exploded and antiquated system of a science acknowledged to be useful, or a science universally believed to be a mere useless and pedantic heap of sophistry and non-sense. (5.1.78)
If education were governed according to supply and demand, the world would have a lot fewer Latin teachers. That's because people would get to choose what they wanted to learn about instead of having the government tell them what to learn.